Er, the last time I arrived in TW without a visa I was granted visa-free entry. I've never met anyone who was able to show me a 'landing visa' in their passport.
Visa free = no visa.
You can't convert nothing into a residence permit. You need a visa to start with, which has to be applied for overseas. The rules may have changed, but you would be wise to confirm this before trusting me or anyone else.
The immigration website makes no mention of landing visas. It does say:
immigration.gov.tw//immig_en ... .asp?id=16
immigration.gov.tw//immig_en ... .asp?id=15
immigration.gov.tw//immig_en ... r.asp?id=9
There is no information on how to obtain a resident visa, naturally, but I'm pretty sure that if you turn up without any kind of visa whatsoever then you won't get one.
EDIT: I found the bit where someone converted a VISA-EXEMPT ENTRY to a visitor visa, and then went on to apply for an ARC without doing a visa run. The answer to the OP's question seems to be that it is possible to arrive without a visa and then obtain an ARC without leaving the country. There still appears to be no such thing as a landing visa, all the same.
SatTV is a Taiwan Citizen, not someone who has had to deal with the realities of being a foreigner in Taiwan in the recent past.
As this question, and the phrase 'landing visa,' seem to come up every other week, maybe someone could make a sticky dealing with this topic?